Charge radii can be used as sensitive indicators to encode information about nuclear structure phenomena, and are influenced by various mechanisms. In this work, charge radii of nuclei with proton number $Z=40$ isotopes are systematically investigated based on the relativistic mean field model using the effective Lagrangian ${\mathrm{NL}3}^{*}$ and PK1 forces. The modified charge radii formula with consideration of the correction deduced from the neutron- and proton-pairs condensation around the Fermi surface is used to capture the local variations of nuclear size. The potential energy surfaces of even-even ${}^{80–110}\mathrm{Zr}$ isotopes are depicted by constraining the axially symmetric quadrupole and hexadecapole moments. The calculated results suggest that the characteristic nuclear shape coexistence phenomena can be clearly observed around the neutron number $N=60$ isotopes. The binding energy differences between the prolate and oblate minima and the systematic evolution of the quadrupole deformation parameters ${\beta }_{20}$ are analyzed as well. Combining the existing literature, the shape-phase transition around $N=60$ is identified from the rapid increase of charge radii, and the nuclei tend to keep prolate shape beyond $N=60$. This means that charge radii play an indispensable role in distinguishing the shape orientation of atomic nuclei. The modified charge radii formula can improve the description of the evolutionary trend of changes of charge radii. An inverted paraboliclike shape of charge radii between the neutron numbers $N=56$ and 68 is predicted. Meanwhile, a rapid increase in charge radii can be predicted across $N=68$.

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• Nuclear radii
• Nuclear shapes and moments
• Nuclear structure & decays
• 90 ≤ A ≤ 149